Saturday, April 19, 2008

Running in the Rain: The Math

I always wanted to knuckle through this problem. So riding back from Florida I worked through the beginnings of it. Presented here is a preliminary mathematical analysis of whether it is better to walk or run in the rain.

It may seem natural that you would want to run in the rain and get through it as fast as you can, but the argument against it is that as you run the rain hits your forward facing area (vs. just hitting you on your upward facing area) thus making more area for the rain to hit. That way you get wetter. We will see how this idea pans out mathematically.

There are multiple ways to tackle this problem. I first started out with a continuum/flux analysis, but I found that since the raindrops are rather discreet points that it didn't make much intuitive sense to deal with them in this manner. Instead I looked at the problem in a different way.

In the end we are concerned with how many raindrops hit our body as we traveled from point A to point B. This would somehow correlate (not necessarily linearly) with how wet we got. We begin by looking at a field of raindrops. Shown above is a simple diagram of a snapshot in time of a field of raindrops. I am making the assumption for this initial study that they are falling straight down. What we need to find out is which of these raindrops we will intersect with. As we travel forward in space and time the raindrops will fall, the slope of the red line reflects upon this fact. As we travel forward, raindrops which once were higher will fall into our path where they will hit us. The slope of the red line is given by:

We can see the effect of this slope by looking at two situations: one if we travel really fast and another if we travel really slowly.

If we travel really fast so that Vperson >> Vrain we would essentially just chop out a section of raindrops. This area (D*h) times the # of raindrops per area (rhorain times your width) yields the total number of raindrops struck.

If instead we walk very very slowly we would expect that the area created by the arrows to be very large and as Vperson -> 0, we should be hit by an infinite amount of rain (at that point you are just standing in the rain.)

After we run the calculations for all situations we end up with:

Where D is the distance traveled, rhorain is the density of rain times your shoulder to shoulder width, w is your depth, front to back and h is your height. A simple equation, we can see that as Vme -> infinity then Rain Hit = rhorain*D*h. Also, if Vme -> 0 then Rain Hit -> infinity, just as we predicted.

The real interesting part is seen when the rain is falling straight down. As mentioned before, one would presume that when you run the rain comes at you at a larger angle and therefore you get struck by more rain, however the math shows that no matter what speed you travel the same number of rain drops will strike your forward facing surface. Fascinating.

To add complication, we are also interested when there are cross winds. Thus the rain isn't falling straight down.

The same idea follows, solve for the slopes and figure out the area of rain drops cut out. I'll spare you the math (I'll add a .pdf with all the details later.) In the end we get,

And there we have it. We look at the limits, as Vme -> inf, then Rain Hit = rhorain*D*h, like before. If Vrain_across=0, we get the same equation as before, which is correct. Also, if Vrain_across=Vme then we only get rain on our head, also correct. And what is the net conclusion from this equation… it is always better to run faster through the rain, no matter what. Below is a contour plot showing variations in your number of rain particles hit (red=lots of rain, blue = little rain) vs. horizontal rain velocity and human velocity. For reference walking speed is around 2-4 mph, and downward rain speed varies from 7-18 mph (I used 12 mph for this example.) You can see a minimum line when your speed equals that of the rain, this is when you are running with the rain. Also, for any given rain speed, running faster always results in less rain hit. The only exception may be if you are running with the rain and start to run faster than it, you could get slightly wetter, but not significantly.

In Conclusion I want to discuss some of the limitations of this study and address them. Firstly, this assumes you are a rectangle. Admittedly, for the majority of America this may not be true, but for an average person we are not too different from one. Also, the assumption that rain hit = wetness is not necessarily true, if a lot of rain is hitting on top of your head much of it may run off. Also, the relative velocity of the rain hitting you may dictate how well it penetrates your clothing (which is also a significant variable.) So keep in mind this is a limited study, but as all physics goes, starting with a simple model usually gives you significant insight into the problem and although it may not be perfect we still can learn something from it.